Welcome to FlyerTalk, sacha728!
Originally Posted by
sacha728
It was about a KLM operated flights with a delta codeshare.
That much is true, and it is correct that this is the reason OP did not enjoy the first bag free benefit provided by the DL AMEX. The terms say "The first checked bag fee waiver will only be applied on flight segments which originate on a Delta or Delta Connection® carrier when you check-in with Delta for both a Delta marketed and Delta operated flight. Codeshare flights are not eligible." This makes many of the sentences and examples in your original post incorrect...
Incidentally there is a corresponding long-running thread in the DL forum, including a wiki, which interested members may peruse as well:
Delta Amex "First Checked Bag Free" Consolidated Thread